Neimark-Sacker bifurcation is the birth of a closed invariant curve
from a fixed point in dynamical systems with discrete time (iterated maps),
when the fixed point changes stability via a pair of complex eigenvalues with
unit modulus. The bifurcation can be supercritical or subcritical,
resulting in a stable or unstable (within an invariant two-dimensional manifold)
closed invariant curve, respectively. When it happens in the Poincare map of a limit cycle,
the bifurcation generates an invariant two-dimensional torus in the corresponding ODE.

then this map is locally conjugate near the fixed point to the normal form, that can be written using a complex coordinate \(z=y_1 + iy_2\) as
\[
z \mapsto (1 + \beta)e^{i \theta(\beta)}z + c(\beta)z|z|^2 + O(|z|^4)
\ ,\]
where \(y=(y_1,y_2)^T \in {\mathbb R}^2\) and \( \beta \in {\mathbb R}\) is the new parameter; see, for example,
Iooss (1979) or Arnold (1983).

If \(d(0) < 0\ ,\) the normal form has a fixed point at the origin, which is asymptotically stable for \(\beta \leq 0\) (weakly at \(\beta=0\)) and unstable for \(\beta>0\ .\) Moreover, there is a unique and stable closed invariant curve that exists for \(\beta>0\) and has radius \(O(\sqrt{\beta})\ .\) This is a supercritical Neimark-Sacker bifurcation (see Figure 1).

If \(d(0) > 0\ ,\) the fixed point at the origin in the normal form is asymptotically stable for \(\beta<0\) and unstable for \(\beta \geq 0\) (weakly at \(\beta=0\)), while a unique and unstable closed invariant curve of radius \(O(\sqrt{-\beta})\) exists for \(\beta <0\ .\) This is a subcritical Neimark-Sacker bifurcation (see Figure 2).

Note that in some books the wrong nondegeneracy condition \({\rm Re}[c(0)] \neq 0\) is given instead of (NS.3).
Applying this wrong condition, one can draw false conclusions about the direction of birth and stability of the
closed invariant curve.

Figure 3: Closed invariant curve with one stable period-6 cycle and one unstable period-6 cycle shown as fixed points of the 6th-iterated of the map.

The smoothness of the closed invariant curve for a fixed parameter value is only finite -
even if the original map is infinitely-differentiable - but increases if \( \beta \to 0\ .\)
The \(O(|z|^4)\)-terms in the normal form cannot be truncated, since they effect
the orbit structure on the closed invariant curve. When these terms are present and depend generically on
\( {\rm arg}(z)\ ,\) the orbits on the invariant curve can either be all everywhere dense
or there exists only a finite number of periodic orbits. The orbit structure varies with \( \beta \ :\)
Exactly two periodic orbits exist in some open parameter interval but disappear on its borders through the saddle-node bifurcation for maps.
A generic map exhibits near the Neimark-Sacker bifurcation an infinite number of these bifurcations of cycles in the
closed invariant curve, corresponding to the borders of the infinite number of such intervals.

Figure 4: Arnold tongues near the Neimark-Sacker bifurcation.

Let \( \lambda=\lambda_1\) and consider \({\rm Re}\; \lambda \)
and \({\rm Im}\; \lambda \) as new independent parameters. On the plane of
these parameters, the unit circle \(|\lambda| = 1 \) corresponds to the
Neimark-Sacker bifurcation. Staying away from the strong resonances and assuming that
the bifurcation is supercritical, we get a stable closed invariant curve for parameters
\(({\rm Re}\; \lambda, {\rm Im}\; \lambda)\) outside the unit circle.
Parameter regions, in which pairs of periodic orbits on the closed invariant curve
exist, approach the unit circle at all rational points
\[
\lambda =e^{i\theta},\ \ \theta=\frac{2\pi p}{q},
\]
as narrow tongues with the \(O((|\lambda|-1)^{(q-2)/2}\)-width. These
Arnold tongues are bounded by the saddle-node bifurcation curves for the cycles.
Far from the Neimark-Sacker bifurcation, different tongues can intersect. At such parameter
values, the closed invariant curve does not exist and two independent saddle-node bifurcations happen with unrelated remote cycles.

Multi-dimensional Case

In the \(n\)-dimensional case with \(n \geq 3\ ,\) the Jacobian
matrix \(A_0=A(0)\) generically has

a simple pair of critical eigenvalues \(\lambda_{1,2}=e^{\pm i \theta_0}\) such that \( e^{ik\theta_0} \neq 1\) for \(k=1,2,3\ ,\) and \(4\ ,\) as well as

According to the Center Manifold Theorem for maps, there is a family of smooth two-dimensional
invariant manifolds \(W^c_{\alpha}\) near the origin. The \(n\)-dimensional
map restricted on \(W^c_{\alpha}\) is two-dimensional, hence has the normal form above
and demonstrates the described bifurcation.

Torus Bifurcation

Suppose that the Neimark-Sacker bifurcation occurs in the Poincare map of a limit cycle in ODE,
so that the fixed point corresponding to the limit cycle has a pair of simple eigenvalues
\( \mu_{1,2}=e^{\pm i \theta_0} \) and all formulated above genericity conditions hold.
Then a unique two-dimensional invariant torus bifurcates from the cycle, while it changes
stability. The intersection of the torus with the Poincare section corresponds to the
closed invariant curve. The torus bifurcation is sometimes called the secondary Hopf bifurcation.

Strong Resonances

The torus bifurcation of limit cycles was known to Andronov. The first paper
on this bifurcation by Neimark (1959) contained an error: He underestimated the role of the strong resonances and
omitted by mistake the corresponding nondegeneracy conditions. However, these conditions are not
merely technical, since near such resonances more than one closed invariant curve can
appear, or no such curves may exist at all. The first complete proof in the absence of
all strong resonances was given by Sacker (1964), who discovered the bifurcation
independently. For the modern theory of strong resonances as the two-parameter phenomena, see
Arnold (1983).